Adaptive Magnetic Hamiltonian Monte Carlo

Wilson Tsakane Mongwe,Rendani Mbuvha,Tshilidzi Marwala

doi:10.1109/access.2021.3127931

Wilson Tsakane Mongwe, Rendani Mbuvha + Show 1 more

Open Access

https://doi.org/10.1109/access.2021.3127931

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Abstract

Magnetic Hamiltonian Monte Carlo (MHMC) is a Markov Chain Monte Carlo method that expands on Hamiltonian Monte Carlo (HMC) by adding a magnetic field to Hamiltonian dynamics. This magnetic field offers a great deal of flexibility over HMC and encourages more efficient exploration of the target posterior. This results in faster convergence and lower autocorrelations in the generated samples compared to HMC. However, as with HMC, MHMC is sensitive to the user specified trajectory length and step size. Automatically setting the parameters of MHMC is yet to be considered in the literature. In this work, we present the Adaptive MHMC (A-MHMC) algorithm which extends MHMC in that it automatically sets the parameters of MHMC and thus eliminates the need for the user to manually set a trajectory length and step size. The trajectory length adaptation is based on an extension of the No-U-Turn Sampler (NUTS) methodology to incorporate the magnetic field present in MHMC, while the step size is set via dual averaging during the burn-in period. Empirical results based on experiments performed on jump diffusion processes calibrated to real world financial market data, a simulation study using multivariate Gaussian distributions and real world benchmark datasets modelled using Bayesian Logistic Regression show that A-MHMC outperforms MHMC and NUTS on an effective sample size basis. In addition, A-MHMC provides significant relative speed up (up to 40 times) over MHMC and produces similar time normalised effective samples sizes relative to NUTS.

Highlights

Markov Chain Monte Carlo (MCMC) methods are a key inference tool in the inference of probabilistic machine learning models [1, 2]
We present the Adaptive Magnetic Hamiltonian Monte Carlo (MHMC) (A-MHMC) algorithm which extends MHMC in that it automatically sets the parameters of MHMC and eliminates the need for the user to manually set a trajectory length and step size
The trajectory length adaptation is based on an extension of the No-U-Turn Sampler (NUTS) methodology to incorporate the magnetic field present in MHMC, while the step size is set via dual averaging during the burn-in period

Summary

Introduction

Markov Chain Monte Carlo (MCMC) methods are a key inference tool in the inference of probabilistic machine learning models [1, 2]. HMC has been successfully employed in the inference of probabilistic machine learning models and has been applied in various fields including renewable energy, health and cosmology [8, 2, 9, 7, 10, 11, 12, 13, 14, 15, 16, 10, 15, 17] This algorithm is the preferred MCMC method in practice due to its ability to incorporate first-order gradient information about the target posterior distribution. HMC still produces samples with relatively high autocorrelations [11, 7]

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